Multisymmetric polynomials on set-theoretic quiver representations

Abstract

Eventually constant set-valued representations of a quiver are set-theoretic analogues of nilpotent representations. In recent work by Green-Holmes-Im, the authors enumerated eventually constant set-valued representations for equioriented cyclic quivers using the directed matrix-tree theorem. In this paper, we extend this enumeration to finite quivers without sinks for which every vertex is the target of sufficiently long paths. We encode the representations as directed acyclic graphs and introduce a recursive source-removal method for certain classes of directed acyclic graphs. This yields a strictly upper triangular matrix enumerator in the incidence algebra of the subset lattice. To compute the cardinality of the eventually constant representations, we compress this enumerator to a matrix indexed by cardinality vectors, the set-theoretic analogues of dimension vectors. We conclude by specializing the formulas to the Jordan quiver and recovering the multisymmetric generating polynomial for the cyclic quiver without using the matrix-tree theorem.